DAC-board based X-band EPR Spectrometer with Arbitrary Waveform Control
Thomas Kaufmann1†, Timothy J. Keller1†, John M. Franck1†, Ryan P. Barnes1, Steffen J.
Glaser2, John M. Martinis3, Songi Han1,4*
1Department
of Chemistry and Biochemistry, University of California Santa Barbara,
Santa Barbara, CA
2Department of Chemistry, Technische Universität München, Germany
3Department of Physics, University of California Santa Barbara, Santa Barbara, CA
4Department of Chemical Engineering, University of California Santa Barbara, Santa
Barbara, CA
= contributed equally
* = corresponding author, Songi Han, Department of Chemistry and Biochemistry
9510, University of California Santa Barbara, CA, USA, Phone: +1 (805) 893-4858,
Fax: +1 (805) 893-4120 , Email: songi@chem.ucsb.edu
†
Key words:
Pulse electron paramagnetic resonance; EPR; X-band; arbitrary waveform
generation; transfer function; excitation profile
Abstract
We present arbitrary control over a homogenous spin system, demonstrated on a
simple, home-built, electron paramagnetic resonance (EPR) spectrometer operating
at 8–10 GHz (X-band) and controlled by a 1 GHz arbitrary waveform generator
(AWG) with 42 dB (i.e. 14-bit) of dynamic range. Such a spectrometer can be
relatively easily built from a single DAC (digital to analog converter) board with a
modest number of stock components and offers powerful capabilities for automated
digital calibration and correction routines that allow it to generate shaped X-band
pulses with precise amplitude and phase control. It can precisely tailor the
excitation profiles “seen” by the spins in the microwave resonator, based on
feedback calibration with experimental input. We demonstrate the capability to
generate a variety of pulse shapes, including rectangular, triangular, Gaussian, sinc,
and adiabatic rapid passage waveforms. We then show how one can precisely
compensate for the distortion and broadening caused by transmission into the
microwave cavity in order to optimize corrected waveforms that are distinctly
different from the initial, uncorrected waveforms. Specifically, we exploit a narrow
EPR signal whose width is finer than the features of any distortions in order to map
out the response to a short pulse, which, in turn, yields the precise transfer function
of the spectrometer system. This transfer function is found to be consistent for all
pulse shapes in the linear response regime. In addition to allowing precise
waveform shaping capabilities, the spectrometer presented here offers complete
digital control and calibration of the spectrometer that allows one to phase cycle the
pulse phase with 0.007° resolution and to specify the inter-pulse delays and pulse
durations to ≤250 ps resolution. The implications and potential applications of these
capabilities will be discussed.
1. Introduction
Typically, pulsed X-band electron paramagnetic resonance (EPR) performance
suffers from insufficient excitation bandwidths. The EPR spectra of nitroxidelabeled macromolecules typically span ~ MHz when motionally averaged and up
to
MHz in the rigid limit at X-band[1; 2; 3]. Even a state of the art
spectrometer with a 1 kW travelling wave tube (TWT) amplifier yields a 90° time on
the order of 10 ns[4], which corresponds to a bandwidth that can barely excite a 200
MHz bandwidth. Furthermore, the frequency-domain excitation profile of a typical
rectangular pulse is far from uniform over this bandwidth[5].
One can obtain superior control over the spin system, in particular over the
excitation profile, by arbitrarily shaping the amplitude and phase of microwave
pulse waveforms as a function of time. Such arbitrary waveform generation (AWG)
has previously been implemented for specific, customized quantum computing
applications[6; 7]. Also, the NMR literature has extensively shown that arbitrarily
shaped pulses can excite spins over a dramatically wider bandwidth than
rectangular pulses of the same power, provide highly uniform excitation profiles[8],
excite sharper spectral slices[9], as well as permit extraordinary control over even
coupled spin systems[10]. In fact, it could be said that AWG pulse shaping induced a
paradigm shift in NMR, and a similar impact would be expected if AWG pulse
shaping in X-band EPR became widely available, versatile, and precise. Indeed, there
has already been much progress in the last 1-2 years towards developing AWGcapable EPR spectrometers by modifying commercial or other pre-existing
instruments[11; 12; 13].
A flexible, modular platform, as pursued in this development, that seamlessly
combines the capabilities of a standard EPR spectrometer with the extensibility
needed to perform precisely controlled, phase-coherent AWG would permit one to
address a variety of specific issues. The ability to uniformly excite either the entire
spectrum or to cleanly excite portions thereof would – in particular – benefit twodimensional (2D) EPR and experiments involving indirect detection. These include
double electron-electron resonance (DEER) [14; 15], solution-state 2D electronelectron double resonance (2D-ELDOR) [16; 17; 18], and hyperfine sublevel
correlation spectroscopy (HYSCORE) experiments[19; 20]. These three techniques
alone have demonstrated a prominent and growing importance throughout
biochemistry by, respectively: providing a tool for measuring nm-scale distances
and distance distributions, probing questions about the interactions between
proteins and lipid membranes, and allowing one to map out the local solvent
environment in glassy sample. Indeed, recent work has already shown[11] that
AWG capabilities can improve the sensitivity of DEER experiments. More generally,
precise excitation is a key prerequisite to enable efficient transfer through multiple
coherence pathways, such as in double quantum coherence (DQC) distance
measurements, which could provide potential advantages over DEER in terms of
sensitivity, in the ability to probe smaller distances, and in allowing one to extract
new types of geometric information [21; 22; 23]. In practice, such a sensitivity
enhancement requires the ability to excite nearly all spins in a sample – here,
optimized AWG pulses offer a potential route for achieving wider excitation
bandwidths, while relying on amplifiers with a standard (~1 kW) power output.
Meanwhile, uniform broadband saturation would improve dynamic nuclear
polarization techniques, easing data interpretation by achieving quantitative
maximal saturation (
=1) of nitroxide probes under all sample conditions—in
contrast to the present situation where the maximal saturation can vary
dramatically for spin labels freely dissolved in solution vs. attached to polymer
chains or surfaces[24; 25]. To achieve the greatest possible benefits for any of these
techniques, it is important not only to implement AWG capability in the
spectrometer but also to be able to seamlessly and coherently “switch on” this
capability for any or all portions of a standard EPR pulse sequence. Such a seamless
implementation of AWG will ultimately allow for EPR experiments whose pulse
sequence evolve in active feedback with the detected signal, similar to that
previously seen in laser spectroscopy[26], with shaped pulses optimized on-the-fly,
allowing optimal spin rotations even in cases where certain experimental
parameters might not be well characterized.
Earlier work in EPR that sought to overcome the long known limitations of
rectangular pulses involved implementation of composite pulses[27], stochastic
excitation[28], and tailored pulses[29]. These pulses offered more uniform
excitation and improved signal-to-noise when compared to rectangular pulses;
however, they still do not employ fully arbitrary pulse shaping capabilities and
therefore didn’t permit arbitrary control over a spin system. This is only because
AWG capabilities in EPR, especially at X-band frequencies and higher, remain a
relatively new technology. Only within the last few years have arbitrary pulses with
ns amplitude and phase resolution been investigated for X-band EPR. AWG radio
frequency pulses, such as adiabatic rapid passages[30; 31] or Frank sequences[32]
have been shown capable of providing broadband excitation, even with limited
irradiation power. Recent studies have begun to employ microwave AWG pulses and
have shown promise for performing broadband EPR excitation[13], improved
evaluation of dipolar couplings via DEER [11; 12], dead-time reduction in high
resonator quality factor (Q-factor)[33], and ultra-wideband inversion of nitroxide
signals[11]. Despite these advances, relatively few spectrometers, thus far, feature
arbitrary waveform generation capabilities in the X-band (8-12 GHz) frequency
regime [7; 11; 13; 33] and many of those rely on modified commercial instruments
as the main platforms[11; 13]. This strategy offers the enormous benefit of allowing
the user to operate within a familiar software environment, and on hardware that is
already engineered to a high level of precision and verified to be fully functional.
However, AWG spectrometers based on commercial instruments require integration
with the commercial software, decreasing design flexibility at the current state.
A substantial amount of versatility is added if the spectrometer is home-built
to be centered around a DAC (digital to analog converter) board that operates as the
main control unit – a strategy primarily used in quantum computing
applications[34]. This approach is necessary, for instance, if one wishes to
seamlessly implement and update corrections for the resonator bandwidth and/or
for amplifier saturation effects applied to the entire waveform (i.e. to all pulses
simultaneously), or if one wishes to calculate the shape of an analytically specified
waveform on the fly or account for variable pulse rise and fall times when
calculating delays. In this study, we demonstrate that the flexibility of the design as
presented here permits seamless and coherent integration of AWG capabilities into
standard EPR experiments and yields entirely new opportunities. While wideband
excitation has been demonstrated at X-band frequencies, most notably in the recent
work by Spindler et al.[12; 13] and Doll et al.[11], many applications might rely
heavily on optimizing pulses that are insensitive with respect to variations in offset
and/or misset of, respectively, the pulse frequency and amplitude. Therefore, it is
important to explicitly demonstrate that arbitrary shaped pulses can invoke
precisely the desired spin response. In this work, we map out the excitation profile
of arbitrarily shaped pulses as a function of resonance offset. In the linear response
regime (tip angles of <30°), this excitation profile corresponds to the frequency
profile of the pulse (i.e. the Fourier transform of the pulse waveform). By analyzing
the response from the spins, we can thus define a transfer function which should be
independent of pulse shape and which we can subsequently use to correct the shape
of various pulses, as demonstrated here for square, sinc, and Gaussian pulses. We
find that we can control the shape of the pulses, as seen by the spins, with high
fidelity.
2. Experimental
A stand-alone, home-built, AWG X-band EPR spectrometer was designed to easily
and precisely manipulate the spin systems of samples that could be of biological
interest. A 1 GHz digital-to-analog converter (DAC) board serves as the central
control and timing unit of the spectrometer, in addition to fulfilling its primary role
of controlling the amplitude (42 dB dynamic range) and phase (0.007° resolution) of
an arbitrary waveform with 1 ns time resolution. The resulting pulse waveform can
span a 1 GHz bandwidth that far exceeds the
MHz bandwidth of a typical
nitroxide spectrum. A Python-based programming platform interacts with the
spectrometer to easily program phase-coherent pulse sequences that can include
square or arbitrarily shaped pulses with arbitrarily timed effective pulse lengths
and inter-pulse delays.
2.1.
Hardware
Fig. 1: Schematic overview of the pulsed AWG X-band EPR spectrometer. Xband microwave paths, indicated in red, transmit the pulse waveforms and carry
the returning signal via coaxial cables, except for components #11, #13, #14, and
#17, which are X-band waveguide components. The microwave carrier is
generated (#2) and amplified (#3) and then mixed (#6) with two digitally
controlled, quadrature 1 GHz transmit waveforms that are generated by the DAC
board (#4) and amplified by two differential amplifiers (#5) to generate shaped Xband pulses at –10 dBm. The output waveform is filtered (#7), amplified to ~43
dBm (#10), and sent to the resonator (#12). The returning signal is power-limited
(#15), and amplified (#16), before being sent to a heterodyne detector comprising
an IQ mixer (#18) that generates two quadrature 1 GHz intermediate frequency
(IF) waveforms, which are amplified (#19) and detected by the storage
oscilloscope (#20). DC to 1 GHz signal paths are shown in blue and are present in
both the AWG unit and the heterodyne detector. The reference oscillator (#21) is
connected to a distribution amplifier (#22) and supplies a 10 MHz signal, shown in
green, which synchronizes the microwave source, DAC board, and detector.
Appendix A describes the various labeled components in detail.
Fig. 2: Photograph and schematic of the arbitrary waveform generation
(AWG) unit. The figure shows the 1 GHz digital-to-analog converter (DAC)
board and the associated components used to modulate the shaped X-band
waveform and to deliver a TTL timing trigger to the detector. Details are
discussed further in Appendix A.2.
The YIG-tuned oscillator (Fig. 1 – #2) outputs a coherent microwave carrier
between 8 and 10 GHz at 10 dBm. As described in Appendix A.1, the output
frequency of this carrier is computer-controlled. This X-band carrier wave next
passes through a preamplifier (MiniCircuits ZX60-183+, Fig 1 – #3, Appendix A.1),
reaching a power of 18 dBm before passing through an IQ mixer (Marki Microwave
IQ-0618 L XP, Fig. 1 – #6) that shapes the amplitude and phase of the microwave
waveform, which leaves the mixer at a power of about –10 dBm. The time-varied
voltage (up to 10 to 13 dBm) modulation across the two quadrature IF (i.e.
intermediate frequency) ports of the IQ mixer that define the arbitrary pulse shapes
are controlled by the DAC board (Appendix A.2) with 1 ns time resolution, i.e. a 1
GHz bandwidth, (see Fig. 1 – #4 and Fig. 2). Thus, the FT of the time-domain pulse
shape, i.e. the frequency profile of the waveform exiting the IQ mixer, has an
arbitrarily controlled profile in frequency space that spans
MHz relative to the
frequency of the carrier wave. Equivalently, the spectrometer controls amplitude
and phase of the X-band pulse sequence with 1 ns time resolution.
Next, the microwave pulses are pre-amplified by 24 dB to 14 dBm (MiniCircuits
ZX60-183+, Fig. 1 – #10) and further amplified by a parallel array of solid-state
amplifiers (Advanced Microwave PA2803-24, Fig. 1 – #10, Appendix A.3) to a peak
amplitude of 41 dBm (12-13 W), before travelling through a circulator (Cascade
Research X-43-2, Fig. 1 – #11) into the microwave resonator (Bruker BioSpin
ER4118X-MD5, Fig. 1 – #12). For all experiments here, the resonator was kept at
room temperature in a static magnetic field of ~0.35 T (Fig. 1 – #1, Appendix A.1).
The circulator then directs the returning EPR signal, along with the resonator
ringdown (Fig. 1 – #11), towards the heterodyne detector (Fig. 1 – #14 to 20). The
quadrature heterodyne detector (Fig. 1 – #14 to 20), is a standard setup based on
Rinard et al.[35], that digitizes the signal on a digital storage oscilloscope with a 1
GHz bandwidth (Agilent Technologies MSO7104B, Fig. 1 – #20). The low noise
amplifier (Fig. 1 - #16) is protected from high power microwave reflections by a
limiter (Fig. 1 - #15) which can tolerate short pulses from a 1kW TWT amplifier. The
recovery time of the LNA and IQ mixer are short compared to the deadtime of the
spectrometer. More details are described in Appendix A.4.
In order to manipulate and detect coherent waveforms with a ~1° accuracy in the
microwave X-band phase, the system needs to consistently synchronize every
hardware component that generates or detects any pulsed waveforms so that the
timing between the leading edge of any trigger pulses and the crests/nodes of any
carrier waves are consistent to within a jitter of ~0.3 ps (see Appendix A.1. for
phase noise values). An oven-controlled 10 MHz oscillator (Electronic Research Co.
Model 130, Fig. 1 – #21) generates a clock signal and an amplifier (Stanford
Research Systems Model FS735, Fig. 1 – #22) duplicate and distributes it to the DAC
board (High Speed Circuit Consultants, software build 10, Fig. 1 – #4), digital storage
oscilloscope (Agilent Technologies MSO7104B, Fig. 1 – #20), and YIG-tuned
oscillator (MicroLambda MLSL-1178, Fig. 1 – #2). Each of the three components
synchronizes to the 10 MHz clock signal via a phase-locked-loop (PLL) mechanism.
The components used to construct the spectrometer have various imperfections.
Imbalances in the DAC board output levels (see Fig. 2) and imbalances in the
amplitude and phase characteristics of the transmit IQ mixer (Fig. 1 – #6) lead to
systematic imperfections in the amplitude and phase of the transmitted pulse
waveforms, as well as a low-level bleed-through (leakage) of the carrier wave.
Similar imbalances in the heterodyne detector’s IQ mixer (Fig. 1 – #18) and
amplifiers (Fig. 1 – #19) can lead to quadrature imbalance or DC offset in the
detected signal. Rather than pursuing the task of improving the performance of
each relevant hardware component, we simply implement digital calibration
routines that correct for these imperfections, a strategy previously implemented by
Martinis et. al.[36] for experiments on superconducting qubits. Such calibration
routines require knowledge of the exact amplitude and phase of the waveform
output by the AWG (i.e. output from the transmitting IQ mixer, Fig. 1 – #6). In order
to avoid detector imperfections coming from the mixers, diodes, or amplifiers that
constitute the home-built heterodyne detector (Fig. 1 – #14 to 20), we employ a
sampling oscilloscope (Fig. 1 – #9), which acquires only one sample point per
acquisition, but does so accurately, linearly, and with a very high (20 GHz) detection
bandwidth. We stroboscopically reconstruct the amplitude and phase of the
transmitting microwave waveform by feeding the sampling scope a phase-coherent
(i.e. consistent to within <0.3 ps) trigger from the DAC board and an unmodulated
reference signal from the microwave carrier, as described in detail in Appendix D.
With knowledge of the exact microwave waveform that is transmitted, we can then
calibrate for imperfections in the AWG and (by using the transmitted waveform as a
known test signal) the detector, as described in Appendix C.2. While the output
transfer function of the AWG system might not be significant relative to the input
transfer function of the resonator (which we correct for with experiments shown in
the Results section), proper calibration for the transmitter IQ mixer’s quadrature
imbalance can prove crucial to one’s ability to properly and easily control the pulse
waveforms. This is because such imbalances are non-linear – i.e. they cannot be
expressed as a matrix transformation (e.g., a scaled Fourier transform) in the
complex waveform space. The Python libraries can automatically include the results
of these calibrations when generating output pulse sequences or detecting the free
induction decay (FID) and echo signals.
In fact, it is our core strategy that the AWG EPR spectrometer is controlled entirely
through in-house Python libraries (see Appendix B.1). These allow the user to
interact with the spectrometer through simple pulse programming commands that
can specify the position of the various delays and timing triggers with a minimal
amount of coding. The shape of the pulse waveforms can be specified either
numerically or by simply writing out their analytical forms (see Appendix B.2). As
the results section will demonstrate, there also is a distinct benefit to specifying the
pulse sequence at an arbitrarily high resolution and allowing the Python library to
automatically down-sample it to the 1 ns resolution of the DAC board (see Appendix
C.1 for details). Similarly, a single function call allows the user to adjust the
acquisition length or sampling time and to acquire signal.
2.2.
Samples
The experiments presented here employed one of two samples, both of which were
loaded into a 3 mm i.d. 4 mm o.d. fused quartz EPR tube (Part 3x4, Technical Glass
Products, Painesville Twp., OH).
Sample A: Several flakes (~1-5 mg) of a solid BDPA (α,γ-Bisdiphenylene-βphenylallyl) complexed with benzene (1:1) were used without further modification
(Product 152560, Sigma Aldrich, St. Louis, MO). Both, the T1 and Tm relaxation times
for this sample are approximately 100 ns[37].
Sample B: Following the procedure used by Maly et. al.[37], BDPA was dissolved in
toluene and then diluted in a polystyrene matrix to a concentration of 46 mmol
BDPA / kg of polystyrene (Product 331651, Sigma Aldrich, St. Louis, MO). The
BDPA-polystyrene-toluene paste was then dried by spreading it onto a watch glass.
After sitting for 6 hours, the sample was dried under vacuum overnight for 12
hours. Then, 28 mg of the powdery, dilute sample was packed into the EPR tube.
The T1 relaxation time, determined by inversion recovery, was ~5 µs, and the phase
memory time, Tm, determined by spin-echo measurements, was ~500 ns.
3. Results and Discussion
In this section, we characterize the AWG X-band EPR spectrometer and demonstrate
the ability to obtain the desired spin response for a given pulse. We first (Section
3.1) perform four tests to verify that this digitally calibrated single-channel
spectrometer allows one to perform standard EPR experiments: (1) We characterize
the phase stability and noise of the instrument. (2) By measuring the fidelity of the
pulses amplified by a solid-state amplifier, we judge the spectrometer’s ability to
generate shaped microwave pulses at higher power. (3) Despite the absence of
traditional phase-shifter components, we successfully implement standard 16-step
(echo sequence) and 4-step (FID sequence) phase cycles to remove experimental
artifacts, thus demonstrating precise control over the phase of our digitally
calibrated pulses. (4) We perform a standard
relaxation measurement to test the
inter-pulse spatial resolution achieved with the DAC board. Finally (Section 3.2), by
observing the spin response over a range of resonance offsets, we obtain the
frequency profile of the pulse waveforms that are seen by the spins. This allows us
to correct for the response profile of the spectrometer in order to generate
rectangular, Gaussian, and truncated sinc pulses with very high fidelity.
3.1.
Verification of Basic Spectrometer Performance
To show that the various hardware components remain phase coherent, EPR signal
was detected (on Sample A) with a varying number of averages, . With perfect
phase coherence, the SNR would scale linearly with . A π/2 rectangular pulse of
60 ns length was repeated times at a rate of 100 kHz, yielding FIDs that were
signal averaged on the oscilloscope and Fourier transformed to yield a spectrum.
The SNR was determined from the integrated signal peak divided by the standard
deviation of the noise. The SNR was re-determined for values of ranging from 500
to 29,500 and plotted against
as shown in Fig. 3. The linear dependence of the
SNR on
shows that the transmitter and receiver systems maintain excellent
phase coherence through the course of many signal averages. Thus, the standard
oven controlled oscillator that we use for our experiments has the stability
necessary to perform standard pulsed EPR experiments for biochemical
applications, if necessary over long timescales required to secure sufficient signal to
noise.
Fig. 3: Signal coherence test. SNR, calculated as explained in the text, as a
function of , where is the number of scans that are averaged.
The absolute SNR reported in Fig. 3 is not optimized to the level reported for other
homebuilt or commercial X-band instruments (SNR = ~25 from 1 scan on
total spins in a 1kGy gamma irradiated quartz sample[38]). However, the reasons
for this are well understood. First, we have not yet implemented a switch to close off
the pre-amplifier (20 dB gain) and solid-state amplifier (36 dB gain), which thus
presently transmit relatively high power noise to the detector. By adding a switch
we expect to reduce the noise power density from the measured value of
W/Hz (-124 dBm/Hz) to approximately the level of thermal noise of the
resonator (
W/Hz, i.e. -174 dBm/Hz, at 300 K), and thus improve the
SNR (calculated as
) by ~
-fold. The loss from the
resonator to the LNA in the detector is approximately 3dB, which could be
optimized. Also, the noise figure of our detection system itself (10 dB degradation in
the SNR) is much higher than expected (2.51 dB) from the design previously
published in [22], corresponding to another 2.4-fold gain in the (
) SNR.
(A systematic diagnosis of the detection system determined that the first amplifier
in the receiver train has a noise figure higher than specifications, and work is
currently underway to resolve this.)
Next, we test the ease and fidelity with which our spectrometer generates a complex
pulse sequence at a moderately high output power. We defined waveforms
analytically at an arbitrarily high resolution (Appendix B.2) to describe a composite
pulse (+x, –x, +y, –y), a triangular pulse, a Gaussian pulse, a truncated sinc pulse, and
an adiabatic rapid passage sech/tanh pulse[31; 39]. The Python libraries
automatically calibrate (Appendix C.2) and down-sample the waveforms to 1 ns
resolution to generate suitable instructions for the DAC board (Appendix C.1). The
pulse sequence was captured on the sampling oscilloscope with absolute phase
information after being amplified to a power of 12-13 W (41 dBm) and further
calibrated using the procedure described in Appendix D. Slight deviations in pulse
amplitude and phase (Fig. 4) can be seen when exceeding 70% of the maximum
output power of the AWG, but are negligible at lower powers. The amplitude and
phase offsets are caused by amplifier saturation which, in principle, can be
corrected for separately from resonator distortions. We therefore expect that we are
not limited by non-linearities present even in higher power amplifiers such as
TWTs.
Fig. 4: Captured pulse sequence with multiple shaped pulses. Real (a)
and imaginary (b) components of a targeted waveform (black dashed lines)
are compared to the microwave waveform captured on the sampling
oscilloscope (grey, solid lines) following the procedures described in
Appendices C.1, C.2, and D. The target waveform consists of a sequence of
composite pulses, a triangular pulse, a Gaussian pulse, a truncated sinc pulse,
and a sech/tanh adiabatic rapid passage (left to right, respectively).
Next we test the ability of this spectrometer to perform standard phase cycling
sequences, which are essential for pulsed EPR experiments due to their ability to
select signal belonging to particular coherence transfer pathways and their ability to
correct for receiver imbalances and other imperfections[40]. Some applications, in
particular double quantum coherence (DQC)-based distance measurements[41],
demand highly accurate phase cycling capabilities. This is typically achieved by
directing the microwave pulse waveform through one of several channels with pre-
set phase delays, e.g. requiring 4 channels to cycle between +x, –x, +y and –y phases.
However, the spectrometer presented here implements a single channel with fully
functional phase cycling that is achieved simply by adjusting the relative scaling in
the two AWG quadrature inputs. Though not utilized here, this scheme offers the
possibility to exploit arbitrary phase values if needed.
A Hahn echo sequence (Fig. 5(a)), with a 60 ns π/2-pulse and a 120 ns π-pulse ( of
1000 ns), was employed to acquire signal from Sample B at a resonator Q of 1,000.
In order to isolate the echo signal without cycling the phase of the pulses and
receiver, one must acquire an off-resonance “background scan” that is subtracted
from an on-resonance signal. We performed such an experiment for comparison.
Then, we carried out experiments using a 16-step phase cycle that selects the
coherence transfer pathway of the echo (Fig. 5(b)) to remove any spurious pulse
ring-down signal and DC offset, as well as unwanted residual FID signal, from the
second pulse without the need for a background scan. As expected and shown by
Fig. 5(c), both background subtraction and phase cycling yield clean, identical echo
signals, while the SNR of the phase cycled result is approximately twice as large
when the same number of total scans are used for both methods. This SNR
improvement is expected since half the scans for the background subtraction
experiment were acquired off-resonance, resulting in half the signal amplitude.
Fig. 5: Hahn echo measurement verifies functionality of AWG-based
phase cycling. Hahn echo pulse sequence (a) and respective coherence
transfer pathway (b), as well as comparison of the signal (c) retrieved with
off-resonance background subtraction (red) vs. a 16-step phase cycle (black)
appropriate for the coherence pathway shown in (b). A 400 MHz digital
band-pass filter was applied to the signal shown in (c). The SNR (integrated
echo intensity vs. RMSD of the noise) is 9.6 for off-resonance background
subtraction and 18.4 for phase cycling.
In the case where we are collecting an FID, phase cycling no longer offers the same
advantage in removing the ring-down from the signal. However, it is well known
that phase cycling can effectively eliminate instrumental imperfections. For
instance, if the two quadrature detection channels are not precisely 90° out of phase
with each other or have different sensitivities, a negative frequency mirror peak
appears in the Fourier transform of the FID[42]. As shown in Fig. 6, when an FID
signal is acquired with a 4-step phase cycle, the phase cycling provided by the
spectrometer successfully removes the artifactual mirror peak that occurs in the
Fourier transform of the FID. Along with the previously demonstrated selection of
the echo signal, this demonstrates the precision and stability of the mixer-based,
digital AWG phase cycling method, which does not require separate pre-set phase
delay channels.
Fig. 6: FID measurement verifies functionality of AWG-based phase
cycling through removal of receiver imbalance artifacts. FID pulse
sequence (a) and coherence transfer pathway (b). Fourier-transform of FID
with magnetic field set 2.4 G off-resonance (c). The negative frequency peak
due to receiver imbalances is successfully eliminated with a digitally
calibrated, mixer-based phase cycle.
Many home-built spectrometers offer a timing resolution significantly better than 1
ns[11; 43]. Such high timing resolution might be needed in the currently presented
spectrometer for select experiments, especially after upgrading the amplifier to a
high power TWT. Since the fundamental timing resolution of the AWG presented
here is only 1 ns, a concern might arise over how well this spectrometer can control
inter-pulse spacing or calibrate the pulse lengths with sufficient precision. However,
while the 1 ns time resolution of the DAC board fundamentally limits the bandwidth
of the pulses in frequency space (and therefore limits the sharpness of the edges of
the pulses), the pulse lengths and inter-pulse delays can still be defined with a time
resolution finer than 1 ns. This can be accomplished by defining the desired
waveform at an resolution, such as 10 ps, where the inter-pulse delays and lengths
are defined with a resolution that is intentionally higher than that achievable with
the DAC board. This high resolution waveform is then convolved with a 1 ns
Gaussian, filtering out frequency components outside the bandwidth of the DAC
board, while preserving a high dynamic range that finely defines the pulse
amplitude. Then, the pulse is converted to the resolution of the DAC board by
reducing the bandwidth in frequency space to 1 GHz to match the bandwidth of the
DAC board.
This “down-sampling” procedure makes use of the large dynamic range of the DAC
board to preserve the distance between the centers (or edges) of the pulses with a
resolution of better than 250 ps, as Fig. 7 experimentally demonstrates.
Fig. 7: Pulse sequence showing delays that are defined with ≤250 ps
time resolution. The pulse sequence (a) consists of two 2 ns wide pulses
with a varying inter-pulse delay of 6 ns, 6.25 ns, 6.5 ns, and 7 ns and
generates a microwave waveform (b) that is captured on the sampling
oscilloscope (as described in Appendix D) and digitally filtered with a 1 GHz
bandpass. As the higher resolution inset (c) shows, delays with ≤250 ps
resolution are generated with high fidelity.
To conclude this section, we present a proof-of-principle EPR experiment that
implements both sub-ns changes in delay times and phase cycling. We employ a
phase-cycled Hahn echo pulse sequence (as in Fig. 6(a-b)) to acquire signal from
Sample A at a resonator Q of 500 and
length of 60 ns. Here, the delay time, ,
between the π/2- and the π-pulse was varied to carry out measurements of phase
decoherence (i.e. measurement of TM) and signal was acquired with 5,000 averages
(Fig. 8). The best-fit exponential decay determined the Tm relaxation time to be
490±14 ns. Fig. 8 (inset) shows the same measurement at a 250 ps resolution in .
This higher resolution data smoothly interpolates between the lower resolution
points, closely following the best-fit exponential decay. We thus find that AWGbased systems offer the benefit of observing the spin-echo relaxation decay with
nearly unprecedented time resolution. This should allow one to determine subtle
features in Tm, such as those coming from multiple overlapping relaxation
contributions [37], with high precision and also to resolve short decay times.
Although we are limited by the rising and falling edges when generating short
pulses, we can specify the length of a pulse to within 250 ps by our down-sampling
procedure, in a similar way to finely tuning the delay length. In situations where
longer pulses are required, such as matched pulses in ESEEM which have been
shown to increase modulation depth by an even order of magnitude in some
cases[44], we can provide precise tip angles in the case of constant B1 amplitude.
Fig. 8: Tm relaxation curve with 250 ps step size. This figure presents a Tm
relaxation curve recorded from a Hahn echo sequence on Sample B. A 12
MHz digital bandpass filter was applied to the data and the echo area was
integrated and plotted against the delay time, , between the pulses, which
was varied in 34 ns steps. The solid black line (in both the main plot and
inset) gives the exponential fit of this data, with Tm = 490±14 ns. The inset
shows the same measurement, acquired with higher time resolution; for this
data, a 250 ps step size was used for the pulse delay.
3.2.
Spin Response to Transfer Function Corrected Pulses
The AWG capabilities of the spectrometer presented here allow us to generate any
arbitrary pulse shape as part of a coherent pulse sequence. However, the pulse
waveform that acts on the spins inside the cavity does not exactly match that leaving
the AWG. In previous studies, a pickup coil has been used to monitor the microwave
waveform inside the cavity. However, the pickup coil can introduce unknown
spatial distortions to the microwaves inside the cavity and also subjects the
measured waveform to its own transfer function, which may not be consistent for all
setups. Therefore, we instead directly monitor the spin response from a sample
with a narrow linewidth, which selects the amplitude of a narrow spectral slice of
the excitation profile of the pulse. By sweeping the field, we map out the excitation
profile of the pulse as a function of resonance offset. This technique allows us to
verify and optimize precise control of the waveform seen by the spins.
The pulse is convolved in the time-domain with an impulse response function that
depends on the resonator -factor, residual miscalibration of the final AWG output,
and additional group delays and imperfections from the transmission lines (to name
some of the most obvious sources):
(Eqn. 1)
Here
is the waveform of the programmed pulse,
is the waveform of the
pulse inside the resonator, and
is the impulse response function. In the
frequency domain, the convolution above is equivalent to a multiplication, i.e.:
(Eqn. 2)
or
where the functions
, respectively.
,
, and
(Eqn. 3)
, are Fourier transforms of
,
, and
We directly obtain the experimental excitation profile,
, by reading out the spin
response in the linear response (i.e. small tip angle) limit. We can then determine
the frequency-domain transfer function,
, of the spectrometer system as
(Eqn. 4)
We can then define a corrected pulse,
, whose Fourier transform is
,
(Eqn. 5)
which experimentally yields the frequency response of the desired waveform, i.e.
, where
is the frequency profile seen by the spins:
(Eqn. 6)
Thus, when the AWG generates the corrected pulse,
, the waveform inside the
cavity,
, will now match the target waveform,
. Thus, by
determining and correcting for the transfer function,
, one can arbitrarily
control the waveform,
, seen by the spins inside the resonator. Note that
the transfer function, as defined above, would also include distortions due to
frequency-dependent detection sensitivity, therefore, for greater accuracy, we could
include the detector frequency response.
As a proof of concept, we seek to show that this spectrometer can arbitrarily tailor
the excitation profiles of pulses over a 60 MHz bandwidth. To start with, the
transfer function of the spectrometer system must be determined over a broad
bandwidth relative to the corrected pulses. Therefore, the excitation profile of a
short 10 ns rectangular test pulse (~10° tip angle) was determined (Fig. 9 (a)). The
magnetic field was swept over a range of ±35 G (with 1 G resolution) around the
resonant magnetic field, while the FID of a BDPA sample (FWHM = 1.4 G) was
recorded. The FID was Fourier transformed, and the integrated signal amplitude (60
MHz integration width) was plotted against the resonance offset (converted from
field using g = 2.00). To calculate the transfer function,
we divide the
experimental excitation profile,
, by the Fourier transform of the pulse,
(Eqn. 4). Because the Fourier transform of the 10 ns rectangular test pulse first
crosses zero only at
MHz, the excitation profile of the test pulse presents
significant signal amplitude over the 60 MHz bandwidth of this experiment.
Therefore, we can accurately determine the transfer function over the bandwidth of
this experiment. To verify that the transfer function is independent of the pulse
shape, we repeated this procedure with a 50ns truncated sinc pulse as the test pulse
(Fig. 9(b)). As expected, the resulting transfer function was in reasonable agreement
with that from the 10 ns rectangular test pulse.
Fig. 9: Spin response to broadband pulses. The spin response from (a)
10ns square and (b) 50ns truncated sinc pulses used to calculate the
resonator transfer function. The solid lines (–) illustrate the experimental
excitation profiles and the dashed lines ( ) illustrate the Fourier transform
of the output pulse waveforms. The transfer functions – calculated from the
ratio of the experimental excitation profile and Fourier of the pulse
waveform – for (a) and (b) are in good agreement.
We applied the transfer function to correct three different pulse waveforms –
rectangular, Gaussian, and truncated sinc – in order to verify a subsequently
increased control over the spin system. The rectangular pulse was 100 ns long. The
sinc pulse with the same maximum power was truncated after the 2nd side lobe and
was adjusted to a length of 250 ns so that it yielded similar signal amplitude. The
Gaussian pulse was 250 ns long with a FWHM of 88 ns and also featured the same
peak power and signal amplitude. The excitation profiles were mapped out with
sample A as shown before, except the magnetic field was swept over a narrower
range of ±12 G in steps of 0.25 G. When not performing the transfer function
correction, the 24 MHz bandwidth of the resonator (
) attenuates the side
features of the excitation profile (Fig. 10). This is especially apparent for the
rectangular (a) and sinc pulses (c). As expected, once we apply the transfer function
correction (d-f), the detected excitation profile matches the Fourier transform of the
desired pulse with very high fidelity (Fig. 10).
The transfer function corrected excitation profiles shown in Fig. 10 were obtained
by defining an absolute value transfer function and contain no phase information. If
instead, we define a complex transfer function, we can correct both the real and
imaginary parts of the excitation profile – corresponding to the X and Y components
of magnetization in the rotating frame. A square 100 ns pulse was sent to the
microwave resonator, which was overcoupled to a Q of ~500. As indicated by the
excitation profile Fig. 11(b), the pulse is significantly distorted by the bandwidth of
the resonator and by imperfections in the various transmission line components (i.e.
waveguides, circulator, etc.). After the complex transfer function correction, the
corrected pulse is significantly different from the original square pulse Fig. 11(c).
The transfer function correction makes the pulse more broadband, thus introducing
new frequency components that cause oscillations in the waveform of the pulse.
These oscillations also indicate that the transfer function of the resonator is not as
intuitive as the Lorentzian-shaped response function that is predicted by an RLC
circuit-like model. Additionally, the small amplitude peaks at the beginning and end
of the pulse can be accounted for by noting that they help compensate for the finite
rise and fall time inherent to the resonator, i.e. these peaks force the resonator to
charge and de-charge, respectively. Once the complex transfer function is applied to
the original pulse, both the real and imaginary parts of the excitation profile are
significantly improved. Remarkably, despite being initially highly attenuated, after
correction, the off-resonance features are scaled properly, as given by Fig. 11(d),
where they closely match the target excitation profile (100 ns square pulse).
Thus, to support our claim of coherent control over a spin system, we have
demonstrated control over the phase of the excitation profile. In other words, The
DAC board-centered spectrometer can generate arbitrary pulse sequences phase
coherently. Coherent excitation is necessary not only for proper phase cycling, but is
also a prerequisite for transferring polarization through multiple quantum
coherence states. Both of these are fundamental for performing double quantum
coherence distance measurements, which with arbitrarily shaped pulse sequences
can be done in an optimized manner or for any other pulse sequence that relies on
coherence transfer. It should be noted that although we have obtained good
agreement between the target and experimental excitation profiles, these
experiments were done with relatively long pulse lengths and narrow bandwidth
samples, where the experimental opportunities should be fully explored by
transitioning to high-power amplifiers.
Fig. 10: Spin excitation profiles resulting from shaped pulses.
Normalized spin excitation profiles (solid lines) generated by a rectangular
(a), Gaussian (b), and truncated sinc (c) pulses, acquired as described in the
text. The analogous transfer function corrected pulses for (a-c) are shown in
(d-f), respectively. The dashed lines show the Fourier transforms of the
excitation pulses as generated by the AWG, which represent the excitation
profiles of the pulses predicted by the small tip angle approximation (which
is valid for these measurements), whereas the solid lines show the
experimentally measured excitation profile of the spins as a function of
resonance offset.
Fig. 11: Phase sensitive transfer function correction for square pulse.
Black and red lines represent real and imaginary part, respectively. For
excitation profiles, solid (–) and dashed (--) lines represent experimental and
target excitation profiles, respectively. (a) time-domain 100 ns square pulse.
(b) spin response to 100 ns square pulse. (c) complex transfer function
corrected pulse (d) spin response to complex transfer function corrected
pulse. As before, excitation profiles were collected using pulses in the linear
response (i.e., small tip angle) regime.
4. Conclusion
With this work, we demonstrate precise control over the spin system in standard
pulsed EPR experiments – achieved with an X-band AWG EPR spectrometer, entirely
home-built around a DAC platform as the main control unit. In particular, we
precisely shape the offset-dependent response of the spins to follow a variety of
different excitation profiles. The spectrometer presented here thus overcomes the
limitations of rectangular pulses by enabling truly arbitrary waveform capabilities
for pulsed EPR. At the same time, it represents an alternative approach to
spectrometer design. Arbitrary control over the phase and amplitude of the pulses,
as well as the timing of all spectrometer components are combined into one digital
control unit. This provides the technical capability for generating precisely
controlled and optimized coherent pulse sequences, dramatically decreasing the
complexity of the instrument and allowing for a detailed level of control and ability
to further optimize the instrument. A Python-based interface easily masks the
complexity such instrumentation might otherwise introduce, by allowing the user to
program the instrument as though controlling a standard pulse sequence with an
arbitrarily high resolution and permitting the user to specify pulses inline using
analytical descriptions of the relevant functions. Down-sampling to the intrinsic
DAC bandwidth and various calibrations are handled automatically and “behind the
scenes” within the Python-based interface. Thus, we believe this control system
offers significant advantages over what is currently available as state-of-the-art
platforms for pulsed EPR spectrometers.
Similar to other recent work[11; 13; 33], we find that a design based on
mixing a shaped intermediate frequency wave with a higher frequency carrier
proves to be extremely versatile and practical. In particular, we note that the carrier
frequency can be changed without the need to recalculate the entire waveform. This
setup could also be directly adapted to several other frequency bands, including Kband (18-26.5 GHz) and Q-band (33-50 GHz), simply by replacing the X-bandspecific components (the microwave source, mixer, and the microwave resonator
and bridge components). The only limitation for applying the same AWG technology
to much higher frequencies is that the intrinsic 1 GHz bandwidth would no longer
cover the bandwidth of the resonator; however commercial DAC boards that
operate at higher bandwidths are already available[11; 45], and, with more effort,
the design of the present DAC board could be scaled to a faster sampling rate and
subsequently higher bandwidth. Furthermore, even in wideband EPR spectrometers
without resonators or with low-Q resonators, where power becomes the main factor
limiting excitation bandwidth, experiments should substantially benefit from the
enhancement in excitation bandwidth provided by arbitrarily shaped pulses.
This work contains similar developments to those by Spindler et al.[12; 13]
and Doll et al.[11] that have implemented add-on AWG units to pre-existing X-band
EPR spectrometers, and have developed techniques to correct for the response of
the resonator as well as the non-linear response of the TWT amplifier. In this study,
we have investigated the fidelity of the excitation profiles of arbitrarily shaped
pulses. As more advanced AWG EPR experiments are developed, the precision of the
arbitrary pulses will become increasingly important and a necessary consideration
in generating multiple quantum coherence pulse sequences.
Further improvements can capitalize on the advances presented here.
Optimization procedures and on-the-fly feedback techniques can be implemented in
this design to improve the excitation profiles. This work was performed on narrow
bandwidth samples at relatively low microwave powers. Thus, integrating a highpower TWT into this setup is required before AWG capabilities are demonstrated to
become truly advantageous over standard EPR techniques. A high-power TWT
would allow one to implement the same level of high-fidelity control, while exciting
spins over a significantly larger bandwidth. By allowing one to excite significant
signal amplitudes from fast-relaxing samples, it would also allow one to take
advantage of the
ps resolution in pulse length and delay timing to acquire
highly resolved relaxation measurements. This would be of great and direct benefit
to characterize fast relaxing samples (i.e. samples with a short
or ), such as
found with many nitroxide labels or transition metal probes under ambient
conditions, and are relevant for many biological applications. At high power, one
should be able to implement very similar methodologies to those presented here.
The shape of the waveforms seen by the spins can still be directly mapped out in the
linear response regime to verify reproduction of the exact waveform desired. Then
(with appropriate correction for any amplifier nonlinearities), the power could be
scaled up into the regime of nonlinear spin response to generate a variety of
interesting effects[46]. In fact, the sampling-scope detection scheme presented here
should allow one to measure and thus correct for a significant portion of the
nonlinear distortions generated at high powers, near the saturation point of the
TWT, thus easily allowing for the types of nonlinear corrections that have been
implemented by Spindler et al. and Doll et al. [11; 13]. Therefore while offering
flexibility, control, and ease of use at X-band, it is of significant value that this
spectrometer platform design can successfully shape the excitation profiles of the
spins – in fact matching the theoretical profiles with fidelity beyond our
expectations, and thus promising significant improvements for a variety of EPR
measurement techniques.
Acknowledgements
We would like to express our sincere gratitude to Bruker BioSpin and especially
Arthur Heiss, Charles Hanson, Ralph Weber, Peter Höfer, and Robert Dick for their
dedicated and sustained support. We thank Steve Waltman (High Speed Circuit
Consultants) for exceptional technical support and advice. We thank Daniel Sank,
Dr. Yu Chen, Dr. Max Hofheinz, and Dr. Erik Lucero from the Martinis group for
helpful advice pertaining to functioning of the DAC board. This work was funded by
the National Science Foundation IDBR Grant, the National Institute of Health R21
Biomedical Instrument Development Grant, and the CNSI Elings Prize Postdoctoral
Fellowship to J. M. F. S.J. Glaser acknowledges support from the DFG (Gl 203/7-1)
and the Fonds der Chemischen Industrie.
Appendix A: Hardware
A.1.
Magnet and Microwave Source
A schematic overview of the spectrometer is shown in Fig. 1 in the main text. A field
controller (Bruker ER 032 M) controls the 0.35 T static field generated by the
electromagnet (Bruker ER 070, Fig. 1 – #1). A YIG-tuned oscillator (MicroLambda
MLSL-1178, Fig. 1 – #2) generates a low-phase-noise (–53 dBc/Hz at 100 Hz offset,
–60 dBc/Hz at 1 kHz offset, –93 dBc/Hz at 10 kHz offset) +10 dBm cw X-band sine
wave, which is split, with one channel pre-amplified (MiniCircuits ZX60-183+, Fig. 1
– #3) to 18 dBm and fed into the home-built AWG unit (Fig. 2) and the other channel
sent to the heterodyne receiver (Section A.4). Communication with the field
controller occurs through a GPIB connection; a GPIB-Ethernet Controller 1.2 by
Prologix, LLC allows the Python library to communicate with this and all other GPIB
devices through a TCP/IP socket. The microwave source is computer controlled
with a USB interface (FTDI FT245R embedded in the UM245R development board)
together with the FTDI D2XX drivers that allow direct access to the device in the
"bit-bang" mode, which permits one to set the voltage levels on the various pins
simultaneously. Both the static field and microwave frequency can be set from a
Python script, which can also read out the current Hall probe reading and whether
or not the microwave source is currently phase-locked.
A.2.
AWG Unit
The AWG unit is comprised of a two-channel 1 GHz DAC board (High Speed Circuit
Consultants, software build 10, Fig. 1 – #4),[36] four Gaussian filters (High Speed
Circuit Consultants) that reduce output noise, two differential amplifiers (Fig. 1 –
#5) that drive the I and Q channels of an IQ mixer (Marki Microwave IQ-0618, Fig. 1
– #6), a low pass filter (Marki Microwave FLP-1250, Fig. 1 – #7) that suppresses
harmonics generated by the IQ mixer, and a directional coupler (Fig. 1 – #8) that
allows monitoring of the shaped microwave output of the AWG unit with 10 dB of
attenuation on a sampling oscilloscope (Tektronix 11801C, Fig. 1 – #9) with two
sampling heads (Tektronix SD 24, Tektronix SD 22), as described in Appendix D.
Additionally, each of the two differential amplifiers has a secondary output that can
be monitored on a standard digital storage oscilloscope.
The DAC board was developed by John Martinis at UCSB[36] in collaboration with
Steve Waltman (High Speed Circuit Consultants), and features two differential (i.e.
two port) outputs with a 14 bit (i.e. 42 dB) dynamic range and 1 ns time resolution.
The memory of the board allows for storage of up to 16 µs long waveforms.
Additionally, parts of the pulse sequence can be called independently and delays of
arbitrary lengths can be inserted. The bandwidth limitation of the AWG unit is set
by the I and Q inputs of the IQ mixer, which feature a bandwidth of 500 MHz and
thus make the IQ mixer the bandwidth limiting component in the setup. A
significant advantage over commercially available AWGs is that the DAC board
provides four differential ECL trigger outputs that can be utilized to control other
components of the spectrometer. For this spectrometer, an ECL-to-TTL converter
was built (ON Semiconductor ECLSOIC8EVB and MC100ELT21) to open and close
the gate of a TWT amplifier. Another ECL trigger output triggers the sampling
oscilloscope as well as the oscilloscope connected to the heterodyne receiver. Thus,
the DAC board not only controls the phase and shape of the microwave pulses, but
also functions as the central timing unit. The FPGA-based design of the DAC board
also allows for custom modifications of its functionality, if desired.
A.3.
Microwave Transmission
The shaped pulse output of the AWG, approximately –10 dBm coming out of the
mixer, is pre-amplified (MiniCircuits ZX60-183+) by 24 dB and further amplified
(Fig. 1 – #10) by either a 1 kW traveling wave tube (TWT) amplifier (Applied
Systems Engineering Model 117), which we measured to have a gain of 60 dB and a
P1dB of 61 dBm, or an array of four solid-state amplifiers. The four amplifiers
(Advanced Microwave PA2803-24) each yield a gain of 30 dB (P1dB 35 dBm each)
and are adjusted with phase shifters to yield coherent outputs that are combined for
a net gain of 36 dB to reach a maximum power of approximately 12-13 W of power
(41 dBm). Specifically, the array of solid-state amplifiers uses 3 splitters
(MiniCircuits ZX10-2-126) to divide the signal into 4 pathways, 3 phase shifters
(Aeroflex Weinschel 980-4) to adjust the phase of 3 of the pathways to match the
first one, before the signal is fed into the 4 amplifiers, and 4 isolators (UTE
Microwave, CT-5450-OT) to protect the amplifier outputs from reflections before
the 4 pathways are combined (Narda Microwave 4326-4) again. The amplifiers are
powered by a linear power supply (Protek 18020M), mounted on a thick aluminum
block and additionally cooled by computer fans. A 4-port circulator, with port 4
terminated (Cascade Research X-43-2, Fig. 1 – #11), directs the microwave pulses to
the resonator (Bruker ER4118X-MD5, Fig 1. – #12) that holds the sample and
returns the reflected signal to the detector.
For the purposes of monitoring the system, a directional coupler (Fig. 1 – #13)
directs –20 dB of reflected signal to the sampling oscilloscope. A second directional
coupler, placed in front of the YIG-tuned source (Fig. 1 – #2), directs –20 dB of the Xband carrier wave to the second channel of the sampling oscilloscope, where it is
used as a reference to determine the reflected microwave signal with absolute phase
information as outlined in Appendix D.
A.4.
Microwave Detection
The microwave signal returning from the resonator is sent to a 3-port circulator
(Cascade Research X43-10-1, Fig. 1 – #14) and from there to a PIN-diode limiter
(Aeroflex ACLM-4571FC31K, Fig. 1 – #15), which protects the low-noise amplifier
(Miteq AMF-3F-09001000-13-8P-L-HS, Fig. 1 – #16) from damage due to high
power microwaves. The aforementioned circulator (Fig. 1 – #14) directs any
microwave power reflected from the limiter into a high-power terminator (Fig. 1 –
#17).
To obtain heterodyne detection, the amplified signal and the output of the
microwave carrier are input into another IQ mixer (Marki Microwave IQ-0618, Fig. 1
– #18), generating two IF (i.e. intermediate frequency) signals in quadrature, which
have a frequency range between DC and 500 MHz. These signals are amplified by
two amplifiers (MiniCircuits ZFL-500+, Fig. 1 – #19) and detected with a (time-base)
digital storage oscilloscope (Agilent Technologies MSO7104B, Fig. 1 – #20) or a high
speed digitizer card (Agilent Acqiris 1082A).
Appendix B: DAC Board Control Software
B.1.
DAC Board Communication
To control the DAC board from a Python platform, the following software is needed:
• LabRAD Manager
• LabRAD Direct Ethernet Server
• LabRAD GHz FPGA Server
• LabRAD Registry Editor
• Python 2.6 or 2.7
• Pylabrad
• Pyreadline
• Pywin32
• SciPy
• Twisted (www.twistedmatrix.com)
• Numpy
• iPython
• Matplotlib
The LabRAD components, basic scripts to run and control the DAC board as well as
an initial set of registry keys for the LabRAD registry can be acquired by contacting
the authors. To simplify the procedure of using the DAC board, the following
environmental variables should be set:
• LabRADHost = localhost
• LabRADNode = EPR
• LabRADPassword = password
• LabRADPort = 7682
The HDL code used for the DAC board FPGA was version 10, as developed by John
Martinis[36]. The Python scripts used to communicate with the DAC board are inhouse software extensively modified from developments by Daniel Sank from the
Martinis group, and can be acquired by contacting the authors.
B.2.
Waveform Generation
The Python scripts that communicate with the DAC board allow arbitrary
waveforms to be generated on-the-fly. These can be based on analytical
descriptions or specified with arbitrary time resolution. Fig. B1 shows an example:
this pulse sequence generates the waveform shown in Fig. 4 (Section 3.1).
The function “make_highres_waveform(list, resolution)” can be called
with a list of tuples that define (1) the type of pulse, (2) the phase of the pulse, and
(3) the length of the pulse or delay. The first parameter can be "rect"
(rectangular pulse), "delay", or "function" (defined by a function – here a
lambda (i.e. inline) function giving an analytical expression for the pulse waveform).
A rectangular pulse uses the second argument to define the pulse phase, which can
be either "x", "y", "–x", "–y" or any angle between 0 and 360 degrees (Fig. B1,
lines 2 and 3). The third parameter defines the length of a pulse in units of seconds.
In case of a delay, the second parameter defines its length. If the pulse type
"function" is chosen, the second parameter is a function that defines the pulse
shape as a function of time; here we employ lambda (inline) functions to generate
analytically described functions that can be edited on-the-fly. Fig. B1 shows a
triangular pulse (line 5), a Gaussian pulse (line 7), a truncated sinc pulse (line 9) and
an adiabatic rapid passage sech/tanh pulse (line 11) as examples. The parameter
“resolution” defines the time resolution for the pulse sequence – 40 ps in this case –
that can be arbitrarily chosen.
The function “digitize(wave)” (line 14) takes the desired waveform as an
argument, filters it with a 1 GHz Gaussian band-pass and down-samples it to a 1 ns
time resolution. The previously acquired calibration parameters (Appendix C.2)
and transfer function calibration (Appendix C.3) are applied, the waveform is
translated into commands for the DAC board, transferred to the DAC board, and
synthesized as described in Appendix C.1.
Fig. B1: Code that generates a pulse sequence – in this case a composite
pulse comprising 4 rectangular pulses of different phases: a triangular pulse,
a Gaussian pulse, a truncated sinc pulse, and an adiabatic rapid passage
sech/tanh pulse.
The language for pulse programming was kept simple to maximize userfriendliness. Arbitrary pulse shapes can be generated within a fraction of a second,
and simple loops can be set up to perform two dimensional experiments, such as the
T2 experiment and coherence test presented in Sec. 3.1.
Appendix C: Digitization and Synthesizer Calibration
Descriptions of the algorithms used to generate, capture, as well as correct and
calibrate arbitrary waveforms are given in this appendix.
C.1.
Digitization and Waveform Generation
To digitize and generate an arbitrary waveform with the AWG from analytical
descriptions, the desired waveform is defined with arbitrary length and resolution,
then filtered with a 1 GHz Gaussian bandpass filter and down-sampled to a 1 ns time
resolution to match the resolution of the DAC board. The IQ mixer calibration
(Appendix C.2) as well as the transfer function calibration (Appendix C.3) are
applied to take hardware imperfections into account and match the desired
waveform as closely as possible. Subsequently, the waveform is translated into
commands for the DAC board and transferred to the system. The following
describes this procedure:
1. Define a desired waveform of arbitrary length and resolution,
2. Filter with 1 GHz Gaussian:
3. Reduce bandwidth to 1 GHz:
for in 1 ns, 2 ns, …, pulselength
4. Apply IQ mixer calibration (Appendix C.2)
5. Apply transfer function correction (Appendix C.3)
6. Translate to DAC board commands
7. Transfer waveform to DAC board
where “ ” indicates time-domain convolution.
C.2.
IQ Mixer Calibration
The IQ mixer used for waveform synthesis exhibits three correctable imperfections:
1. An amplitude imbalance between the two channels, i.e. “parity imbalance.”
2. The phase angle between the two channels is not necessarily 90 degrees.
3. Microwave leakage occurs when no voltage is applied to both channels.
To correct for all three of these issues, a plane wave
is generated via the DAC
board, mixed with the reference oscillator via the IQ mixer, and the output
waveform
is captured on the sampling oscilloscope. In general, where an input
waveform,
has the form
(Eqn. 7)
where
and
are real numbers giving, respectively, the amplitude and phase
modulation of the waveform, the amplitude and phase of the output waveform will
be altered according to equations
(Eqn. 8)
(Eqn. 9)
as a result of these imperfections[47]. The
and represent the output parity
imbalance, the
and represent the input parity imbalance, the
and denote
a phase shift (arising from slightly different path lengths in each channel), and the
and represent the DC offset of the real (Eqn. 8) and imaginary (Eqn. 9)
components of the output waveform.
Specifically, after outputting a plane-wave
, the real and imaginary components
of the output waveform, which are captured on the sampling scope, are fit to
equations (Eqn. 8) and (Eqn. 9), respectively, as a function of the input phase.
Once these calibration parameters have been determined, one can solve Eqn. 8 and
Eqn. 7 with a standard zero-finding algorithm to determine the input values of
and
needed to yield any desired output amplitude and phase (i.e.
and
angle
, respectively. This corrects the amplitude and phase of the output
waveform. In particular, the microwave leakage is corrected by subtracting the
terms CR and CI from the input waveform’s real and imaginary components,
respectively. This applies a voltage to each channel of the mixer to minimize
microwave leakage at the output and allows for an isolation of more than 50 dB post
amplification.
The IQ mixer used for detection exhibits the same issues as the mixer used for
waveform synthesis. Therefore, an analogous calibration procedure can be applied,
the difference being that non-linearities are accounted for in post processing.
C.3.
Transfer Function Calibration
To maximize the fidelity with which pulses are generated, the transfer function of
the spectrometer can be taken into account as a final step by capturing the
generated waveform and comparing it to the desired pulse shape. Linear response
theory states that the output waveform,
, can be described as a convolution of
the input waveform,
, with the impulse response of the system – i.e. the
inverse Fourier transform of the transfer function,
, of the system. We
represent this as
(Eqn. 10)
where
represents the inverse Fourier transform and again represents a
time-domain covolution. After Fourier transformation, the transfer function in
frequency space can be found, following from
(Eqn. 11)
The transfer function,
, can be subsequently applied to the initial waveform
function to generate the desired waveform
as described in the procedure
below.
Instead of using the desired output pulse to determine the transfer function, a 1 ns
rectangular pulse on each DAC channel can be used to determine the impulse
response function of the spectrometer and the transfer functions for both channels
(corresponding to a real 1 ns pulse) and (corresponding to an imaginary 1 ns
pulse) are added to determine the total transfer function, . The procedure is
described below. This procedure was not utilized in the experiments presented, but
offers opportunity for future applications.
1. Generate 1 ns square pulse
on DAC channel A and capture impulse
response
on sampling oscilloscope
2. Fourier transform the generated waveform to yield
and the
impulse response to yield
3. Find the transfer function
4. Repeat steps 1 through 3 with DAC channel B to find total transfer function
5. Find Fourier transform of corrected output waveform
6. Inverse Fourier transform corrected output waveform to yield
Appendix D: Waveform Capturing
To be able to generate arbitrary waveforms with a defined phase, the absolute phase
of the generated pulse needs to be determined, so that a calibration procedure can
be applied. Any microwave pulse or reflection of the resonator in the system can be
detected with absolute phase information by employing the second channel of the
sampling oscilloscope, which is fed with the attenuated output of the microwave
source. The phase of each data point of the signal channel is compared to the phase
of the constant amplitude reference wave. The procedure is described below, where
gives the Heaviside step function (
for
and
for
.
1. Simultaneously capture plain microwave carrier
analyzed
and waveform to be
2. Generate analytic signal and apply bandpass filter to reduce noise:
,
with
3. Determine absolute phase:
4. Apply bandpass filter to reduce noise:
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Amplitude [a.u.]
(d)
(e)
(f)
Resonance Offset [MHz]
Graphical abstract
•
We present a 1GHz DAC-board based home-built EPR spectrometer
operating at X-band
•
Supported by a fully customizable Python programming platform
•
Highly precise phase (0.007°), amplitude (42dB), and delay (≤250ps)
resolution
•
Demonstrates a transfer function correction to compensate for resonator
bandwidth